Timeline for Connectedness properties of groups of homeomorphisms
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Mar 12, 2014 at 19:50 | vote | accept | Ludolila | ||
| Mar 10, 2014 at 20:11 | comment | added | user46855 | @Ludolila sorry, my poor English made me understand "(compact) or (locally compact and locally connected)" | |
| Mar 10, 2014 at 8:19 | comment | added | Ludolila | @user46855 I took $X$ to be locally compact and locally connected (and Hausdorff). In this case the compact open topology turns $H(X)$ into a topological group. See for example "Topologies for homeomorphisms groups" by Arens. I will check Comfort, though, sounds interesting. | |
| Mar 9, 2014 at 22:49 | comment | added | user46855 | According to Comfort (chap. 24 in handbook of set-theoretic topology, pag. 1242) citing de Groot, the groups of homeomorphism of compact connected spaces are arbitrary groups, and so the discrete topology might be the only group topology. @Ludolila might want stronger conditions on $X$ to have a topological group $H(X)$ | |
| Mar 9, 2014 at 20:24 | history | edited | Ludolila | CC BY-SA 3.0 |
deleted 96 characters in body
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| Mar 9, 2014 at 20:23 | comment | added | Ludolila | @JohnPardon: Just to make sure: if $C(X)$ is a topological vector space, then it will be path connected. So the answer to the question "which topologies make $C(X)$ connected" is "the topologies that make $C(X)$ a topological vector space"? | |
| Mar 9, 2014 at 19:57 | comment | added | Ludolila | @JosephVanName No, I haven't even heard about it. Just finished reading about it in Wikipedia, and it's very interesting! Thanks for the idea! | |
| Mar 9, 2014 at 19:45 | comment | added | John Pardon | @Ludolila: well then $C(X)$ is a vector space and thus contractible. | |
| Mar 9, 2014 at 19:43 | comment | added | Ludolila | @JohnPardon $C(X)$-the space of all continuous functions on $X$. | |
| Mar 9, 2014 at 19:12 | comment | added | Joseph Van Name | @Ludolila. Have you read about the mapping class group? Essentially the mapping class group of a space is the group of autohomeomorphisms modulo the subgroup of autohomeomorphisms isotopic to the identity. In particular, the mapping class group is the collection of all path components in $Aut(X)$. | |
| Mar 9, 2014 at 19:08 | answer | added | Ali Taghavi | timeline score: 1 | |
| Mar 9, 2014 at 19:03 | comment | added | John Pardon | The specific question about connectedness of $H_+(S^1)$ is easy to treat explicitly: you can just write down a path between any homeo and the identity. | |
| Mar 9, 2014 at 19:02 | comment | added | John Pardon | What is $C(X)$? | |
| Mar 9, 2014 at 18:46 | history | asked | Ludolila | CC BY-SA 3.0 |